How to best sharpen eye pencils?

Sharpening eye pencils is the bane of my existence. I love it when you get a new pencil, with its nice sharp tip...but I can never sharpen them to get the same sharp tip thereafter. I think it is the sharpeners. They all seem to contain a little "stop" near the far end of the sharpening hole, which causes the end to be blunt. It is like a little piece of metal stuck in the sharpening hole that prevents the pencil from going too far forward and getting a nice point on it.
I tried putting the pencils in the freezer, but all that did was make the tips break. (I've never had an eye pencil tip break on me while sharpening, so I can only assume that the freezing did it.)
The one I'm using now is Sonia Kashuk, which is made in Germany. Does anyone have any sharpeners that they would recommend? And how often do they need to be replaced?
I tried putting the pencils in the freezer, but all that did was make the tips break. (I've never had an eye pencil tip break on me while sharpening, so I can only assume that the freezing did it.)
The one I'm using now is Sonia Kashuk, which is made in Germany. Does anyone have any sharpeners that they would recommend? And how often do they need to be replaced?
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Low-porosity 2a/b baby-fine hair
RO: IN Harmonic
LI: CJ Beauticurls; Jessicurl (JC) Gelebration Spray
DC: JC Deep Conditioning Treatment; Weleda Rosemary Conditioning Oil
Stylers: JC Confident Coils; IN Volumizing Spray; IN Perfect Hold Hairspray; John Masters Sea Mist; Bumble Bb. Gel; Kenra Firm-Hold Gel; Sevi Blue Yarrow Sea Salt Spray
Low-porosity 2a/b baby-fine hair
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I was told by some people on another forum to try putting the eyeliner pencils in the freezer for 5-10 mins. to try to get them a little harder so that they would sharpen better. But all it did was make them break. And if the problem is the little stopper thingy in the sharpener, I don't know how freezing the pencils would fix the problem. I'll let you know if I find a sharpener that meets my standards!
RO: IN Harmonic
LI: CJ Beauticurls; Jessicurl (JC) Gelebration Spray
DC: JC Deep Conditioning Treatment; Weleda Rosemary Conditioning Oil
Stylers: JC Confident Coils; IN Volumizing Spray; IN Perfect Hold Hairspray; John Masters Sea Mist; Bumble Bb. Gel; Kenra Firm-Hold Gel; Sevi Blue Yarrow Sea Salt Spray
Low-porosity 2a/b baby-fine hair
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RO: IN Harmonic
LI: CJ Beauticurls; Jessicurl (JC) Gelebration Spray
DC: JC Deep Conditioning Treatment; Weleda Rosemary Conditioning Oil
Stylers: JC Confident Coils; IN Volumizing Spray; IN Perfect Hold Hairspray; John Masters Sea Mist; Bumble Bb. Gel; Kenra Firm-Hold Gel; Sevi Blue Yarrow Sea Salt Spray
Low-porosity 2a/b baby-fine hair
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RO: IN Harmonic
LI: CJ Beauticurls; Jessicurl (JC) Gelebration Spray
DC: JC Deep Conditioning Treatment; Weleda Rosemary Conditioning Oil
Stylers: JC Confident Coils; IN Volumizing Spray; IN Perfect Hold Hairspray; John Masters Sea Mist; Bumble Bb. Gel; Kenra Firm-Hold Gel; Sevi Blue Yarrow Sea Salt Spray
Low-porosity 2a/b baby-fine hair